Every term times every term
To expand two brackets, multiply each term in the first by each term in the second, then collect like terms.
- (x + 3)(x + 5)
- x × x = x², x × 5 = 5x, 3 × x = 3x, 3 × 5 = 15
- x² + 5x + 3x + 15
- = x² + 8x + 15
Squared brackets are two brackets: (x + 4)² = (x + 4)(x + 4) = x² + 8x + 16. It is not x² + 16. That missing middle term is the most common lost mark in this topic.
Factorising runs it backwards
Look at the expansion above: 8 = 3 + 5 and 15 = 3 × 5. So to factorise x² + bx + c, find two numbers that multiply to c and add to b.
- x² + 2x − 15
- multiply to −15, add to 2: 5 and −3
- = (x + 5)(x − 3)
Signs: if c is negative, the two numbers have opposite signs. If c is positive, they have the same sign as b. Expand your answer in your head to check.
Two shortcuts
- Difference of two squares: x² − 49 = (x + 7)(x − 7). Two squares with a minus between them.
- Common factor: x² + 6x has no number term, so take x out: x(x + 6). With 3x² + 12x, take 3x out: 3x(x + 4).
Same expression, same graph
Type y = (x + 2)(x + 3) on the first line and y = x² + 5x + 6 on the second. You see one curve, not two: expanding and factorising never change the value, only the way it is written.